Problem: Factor the following expression: $-7$ $x^2+$ $24$ $x$ $-9$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(-9)} &=& 63 \\ {a} + {b} &=& & & {24} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $63$ and add them together. The factors that add up to ${24}$ will be your ${a}$ and ${b}$ When ${a}$ is ${3}$ and ${b}$ is ${21}$ $ \begin{eqnarray} {ab} &=& ({3})({21}) &=& 63 \\ {a} + {b} &=& {3} + {21} &=& 24 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 +{3}x +{21}x {-9} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 +{3}x) + ({21}x {-9}) $ Factor out the common factors: $ x(-7x + 3) - 3(-7x + 3) $ Notice how $(-7x + 3)$ has become a common factor. Factor this out to find the answer. $(-7x + 3)(x - 3)$